Physical quantities and definitions

Hi everybody,
I just want your opinion in some questions: First of all, how do we define a physical quantity? Is it a mathematical creation that helps us describe something happening in nature? And all these formulas that we have come up with: example F=dp/dt . Are they mathematics or physics? When we say that F is force and it's a vector, what do we exactly mean? A vector is mathematical term and Force is a physics term: so we apply all the rules that are used in vectors to do calculations in order to come up with a vector that represents a physical quantity, like force? In other words, all the formulas are plain mathemaics , but the quantities used in them represent quantities in physics, that we have defined in a certain way? To sum up, we have come up with a way of describing physical quantities by numbers, and generally by mathematical creations, so we apply everything that is true for mathematics to these quantities to come up with mathematical formulas that connect the values of physical quantities through a function, example s=(1/2)*g(t^2) is a function in which t represents time and s distance, but in any other aspect is plain mathematics, and we can do anything that applies to mathematics, for example algebraic calculations, and come up with a right result, representing distance or any other quantity?And when we say that F=Gm*m'/(r^2) is the force between two objects with masses m and m' then we actually mean that if we put the value of their masses in this formula, taking the product of them and with consant G and then divide it with the number that represents the product of distance with distance then we come up with a number that represents the force between them?
Are all these right? I just want to make these things clear in my head

Hi everybody,
I just want your opinion in some questions: First of all, how do we define a physical quantity? Is it a mathematical creation that helps us describe something happening in nature? And all these formulas that we have come up with: example F=dp/dt . Are they mathematics or physics? When we say that F is force and it's a vector, what do we exactly mean? A vector is mathematical term and Force is a physics term: so we apply all the rules that are used in vectors to do calculations in order to come up with a vector that represents a physical quantity, like force? In other words, all the formulas are plain mathemaics , but the quantities used in them represent quantities in physics, that we have defined in a certain way? To sum up, we have come up with a way of describing physical quantities by numbers, and generally by mathematical creations, so we apply everything that is true for mathematics to these quantities to come up with mathematical formulas that connect the values of physical quantities through a function, example s=(1/2)*g(t^2) is a function in which t represents time and s distance, but in any other aspect is plain mathematics, and we can do anything that applies to mathematics, for example algebraic calculations, and come up with a right result, representing distance or any other quantity?And when we say that F=Gm*m'/(r^2) is the force between two objects with masses m and m' then we actually mean that if we put the value of their masses in this formula, taking the product of them and with consant G and then divide it with the number that represents the product of distance with distance then we come up with a number that represents the force between them?
Are all these right? I just want to make these things clear in my head

The mathematics embodies abstract relationships and behavior, for example of vectors in a vector space. Particular physical quantities may exhibit the relations and behavior of some defined mathematical system.. For example force behaves like a vector; you can use vector addition to compute the sum of two forces, you can choose a basis and resolve a force into its components in that basis, and so on. In fact you won't find any vector space property that can't be exhibited in the case of forces. So physicists sloppily say a force IS a vector; it would be more careful to say that forces instantiate vectors.

All the formulas are based on thinking about the physical system, such as rate-time-distance, and assigning the mathematical properties that are appropriate; distance = rate * time. Then it is tested in practice. How many RTD propblems do you suppose have been worked by engineers since the time of the ancient Greeks who thought it up? The formula has never failed. So we accept it.

I cant speak for the op, but maybe he was asking if quanities like focre and mass really exist, in that they correspond directly to something real. In other words, is physics isomorphic to reality quantity by quantity, or is the entire system just one of possibly many mathematical formalisms that gives accurate predictions?

Thank you for your answers! They were really helpful. I just want to add that StatusX got to the point i wanted to make:
"maybe he was asking if quantities like force and mass really exist, in that they correspond directly to something real. In other words, is physics isomorphic to reality quantity by quantity, or is the entire system just one of possibly many mathematical formalisms that gives accurate predictions"
That is what i am asking, what are these formulas and what do they represent